\(\int e^{4 \coth ^{-1}(a x)} (c-a^2 c x^2)^3 \, dx\) [583]

Optimal result

Integrand size = 22, antiderivative size = 35 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 (1+a x)^6}{3 a}-\frac {c^3 (1+a x)^7}{7 a} \]

[Out]

1/3*c^3*(a*x+1)^6/a-1/7*c^3*(a*x+1)^7/a

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 45} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 (a x+1)^6}{3 a}-\frac {c^3 (a x+1)^7}{7 a} \]

[In]

Int[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]

[Out]

(c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6275

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^3 \, dx \\ & = c^3 \int (1-a x) (1+a x)^5 \, dx \\ & = c^3 \int \left (2 (1+a x)^5-(1+a x)^6\right ) \, dx \\ & = \frac {c^3 (1+a x)^6}{3 a}-\frac {c^3 (1+a x)^7}{7 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {c^3 (1+a x)^6 (-4+3 a x)}{21 a} \]

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]

[Out]

-1/21*(c^3*(1 + a*x)^6*(-4 + 3*a*x))/a

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29

method	result	size
gosper	\(-\frac {c^{3} x \left (3 a^{6} x^{6}+14 a^{5} x^{5}+21 a^{4} x^{4}-35 a^{2} x^{2}-42 a x -21\right )}{21}\)	\(45\)
default	\(c^{3} \left (-\frac {1}{7} a^{6} x^{7}-\frac {2}{3} a^{5} x^{6}-a^{4} x^{5}+\frac {5}{3} a^{2} x^{3}+2 a \,x^{2}+x \right )\)	\(45\)
risch	\(-\frac {1}{7} a^{6} c^{3} x^{7}-\frac {2}{3} a^{5} c^{3} x^{6}-a^{4} c^{3} x^{5}+\frac {5}{3} a^{2} c^{3} x^{3}+2 a \,c^{3} x^{2}+c^{3} x\)	\(60\)
parallelrisch	\(-\frac {1}{7} a^{6} c^{3} x^{7}-\frac {2}{3} a^{5} c^{3} x^{6}-a^{4} c^{3} x^{5}+\frac {5}{3} a^{2} c^{3} x^{3}+2 a \,c^{3} x^{2}+c^{3} x\)	\(60\)
norman	\(\frac {-c^{3} x +a^{4} c^{3} x^{5}-a \,c^{3} x^{2}+\frac {1}{3} a^{2} c^{3} x^{3}+\frac {5}{3} a^{3} c^{3} x^{4}-\frac {1}{3} a^{5} c^{3} x^{6}-\frac {11}{21} a^{6} c^{3} x^{7}-\frac {1}{7} a^{7} c^{3} x^{8}}{a x -1}\)	\(90\)
meijerg	\(\frac {c^{3} \left (-\frac {a x \left (-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{315 \left (-a x +1\right )}-8 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{3} \left (-\frac {a x \left (-14 a^{5} x^{5}-21 a^{4} x^{4}-35 a^{3} x^{3}-70 a^{2} x^{2}-210 a x +420\right )}{70 \left (-a x +1\right )}-6 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{3} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{3} \left (\frac {a x \left (-20 a^{6} x^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}+\frac {6 c^{3} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}-\frac {6 c^{3} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{3} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{3} x}{-a x +1}\)	\(409\)

[In]

int(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/21*c^3*x*(3*a^6*x^6+14*a^5*x^5+21*a^4*x^4-35*a^2*x^2-42*a*x-21)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \]

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

-1/7*a^6*c^3*x^7 - 2/3*a^5*c^3*x^6 - a^4*c^3*x^5 + 5/3*a^2*c^3*x^3 + 2*a*c^3*x^2 + c^3*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- \frac {a^{6} c^{3} x^{7}}{7} - \frac {2 a^{5} c^{3} x^{6}}{3} - a^{4} c^{3} x^{5} + \frac {5 a^{2} c^{3} x^{3}}{3} + 2 a c^{3} x^{2} + c^{3} x \]

[In]

integrate(1/(a*x-1)**2*(a*x+1)**2*(-a**2*c*x**2+c)**3,x)

[Out]

-a**6*c**3*x**7/7 - 2*a**5*c**3*x**6/3 - a**4*c**3*x**5 + 5*a**2*c**3*x**3/3 + 2*a*c**3*x**2 + c**3*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{7} \, a^{6} c^{3} x^{7} - \frac {2}{3} \, a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} + \frac {5}{3} \, a^{2} c^{3} x^{3} + 2 \, a c^{3} x^{2} + c^{3} x \]

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/7*a^6*c^3*x^7 - 2/3*a^5*c^3*x^6 - a^4*c^3*x^5 + 5/3*a^2*c^3*x^3 + 2*a*c^3*x^2 + c^3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (31) = 62\).

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.23 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {{\left (3 \, c^{3} + \frac {35 \, c^{3}}{a x - 1} + \frac {168 \, c^{3}}{{\left (a x - 1\right )}^{2}} + \frac {420 \, c^{3}}{{\left (a x - 1\right )}^{3}} + \frac {560 \, c^{3}}{{\left (a x - 1\right )}^{4}} + \frac {336 \, c^{3}}{{\left (a x - 1\right )}^{5}}\right )} {\left (a x - 1\right )}^{7}}{21 \, a} \]

[In]

integrate(1/(a*x-1)^2*(a*x+1)^2*(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

-1/21*(3*c^3 + 35*c^3/(a*x - 1) + 168*c^3/(a*x - 1)^2 + 420*c^3/(a*x - 1)^3 + 560*c^3/(a*x - 1)^4 + 336*c^3/(a
*x - 1)^5)*(a*x - 1)^7/a

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {a^6\,c^3\,x^7}{7}-\frac {2\,a^5\,c^3\,x^6}{3}-a^4\,c^3\,x^5+\frac {5\,a^2\,c^3\,x^3}{3}+2\,a\,c^3\,x^2+c^3\,x \]

[In]

int(((c - a^2*c*x^2)^3*(a*x + 1)^2)/(a*x - 1)^2,x)

[Out]

c^3*x + 2*a*c^3*x^2 + (5*a^2*c^3*x^3)/3 - a^4*c^3*x^5 - (2*a^5*c^3*x^6)/3 - (a^6*c^3*x^7)/7